SOLUTION: Find the sum of the geometric series: 1+ sqrt6+ 6+...+7776

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Question 540972: Find the sum of the geometric series:
1+ sqrt6+ 6+...+7776
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
Each term in the series is the previous one multiplied by sqrt%286%29.
It is a geometric series.

I multiplied by %28%28sqrt%286%29%2B1%29%2F%28sqrt%286%29%2B1%29%29=1 to rationalize (get rid of the square root in the denominator).
You may get to
sum=%28%287776sqrt%286%29-1%29%2F%28sqrt%286%29-1%29%29 from a carefully applied formula from your book, or (if allowed) deduce it from
sum*1=1+sqrt6+6+.....+7776, so
sum*sqrt6=sqrt6+6+6sqrt6+.....+7776*sqrt6, so subtracting
sum*sqrt6 - sum*1 = sum*(sqrt6-1)=7776*sqrt6-1
ALTERNATE WORK-UP
7776=6%5E5 We knew it had to be a power of 6.
can be grouped as


and the parentheses can be easily calculated as
1%2B6%2B6%5E2%2B6%5E3%2B6%5E4%2B6%5E5=%286%5E6-1%29%2F%286-1%29=46655%2F5=9331 and
1%2B6%2B6%5E2%2B6%5E3%2B6%5E4=%286%5E5-1%29%2F%286-1%29=7775%2F5=1555